Alberto, Barbara, and Chris each earned money working on a certain job

OFFICIAL EXPLANATION

The words mean and median mean that this problem is testing Statistics. Not much information is given in the question stem so jot down the question and move on to the statements.

Mean > 1.5 median?

(1) INSUFFICIENT: Translate this statement into equations. Assume C is the lowest paid worker, B is next, and A is the highest (it doesn’t matter which variable you assign to which salary). Translate the statement into an inequality.

B – C < 1/3 (A – C) Distribute

B – C < (1/3)A – (1/3)C Add C t both sides

B < (1/3)A + (2/3)C Multiply by 3 to eliminate fractions

3B < A + 2C


Now translate the question into algebra to see if the statement helps.

Mean > 1.5 median?

(A+B+C)/3>1.5B Multiply by 3

A + B + C > 4.5B? Subtract B from both sides

A + C > 3.5B?

It is not clear how the information from the statement would help to answer this question. You could Test Cases at this point to confirm.

Case 1: Pick cases that fit the rephrased statement: 3B < A + 2C. Also, remember the assumptions about the order of pay: A > B > C. Pick a value for B that will be easy to work with and then choose A and C to fit the inequality.

B = 10, A = 25, C = 5

Median = 10 Mean ≈ 13.3

In this case, the answer is No, the mean is not more than 1.5 times the median.

When thinking about Case 2, consider a very large value for the highest salary, holding the other information constant: outliers tend to skew the mean.

Case 2:

B = 10, A = 90, C = 5

Median = 10 Mean = 35. The answer is Yes, the mean is more than 1.5 times the median.

Since both Yes and No are possible answers to the question, this statement is INSUFFICIENT. Eliminate choices (A) and (D).


(2) SUFFICIENT: This statement tells us Alberto makes more than 4 times the second highest paid worker; make that worker B.

A > 4B

What do you know about the mean and median based on this information?

median = B

Mean=(A+B+C)/3 Substitute A with > 4B

Mean=(>4B + B + C)/3

From this statement, there is no upper limit on what Alberto makes (e.g. Alberto could make $1M when the other two people made $5 each); if Alberto makes a very high salary the mean will be very large relative to the median and the answer to the question will be Yes. So the issue is whether you can get an answer of No. Consider what would make the mean as small as possible given the information known: if A was just bigger than 4B and C = 0.

Mean=(4B + B + 0)/3≈1.66B

The smallest possible value for the mean is ~1.66B, which is still larger then 1.5 times the median of B. The answer to the question is always Yes, so this statement is SUFFICIENT. Eliminate choices (C) and (E).

The correct answer is (B).­

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